Discrete form of the Huygens-Fresnel principle: to the multi-dimensional analog of the Nyquist–Shannon sampling theorem

• Published in: International journal of information technology (Singapore. Online) Vol. 15; no. 7; pp. 3751 - 3759
• Main Authors: Vitulyova, Ye S., Suleimenov, I. E., Matrassulova, D. K., Bakirov, A. S.
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.10.2023; Springer Nature B.V
• Subjects: Acoustics; Artificial Intelligence; Computer Imaging; Computer Science; Fourier transforms; Green's functions; Helmholtz equations; Image Processing and Computer Vision; Information theory; Machine Learning; Original Research; Pattern Recognition and Graphics; Principles; Radiation; Sampling; Software Engineering; Theorems; Vision; Wave propagation; Helmholtz equation; Wave processes; Green's function; Information theory
• ISSN: 2511-2104; 2511-2112
• DOI: 10.1007/s41870-023-01423-3

Discrete form of the Huygens-Fresnel principle valid in the case when the factor of inhomogeneous waves can be neglected is proposed. It is shown that the description of the propagation of wave oscillations obeying the Helmholtz equation can be reduced to finding the value of the amplitudes in discrete points corresponding to a certain lattice in this case. Namely, it is possible to pass from the classical Green's functions (fundamental solutions) to the use of finite Green's functions, which are finite everywhere. This approach allows us to solve the same problem that the Nyquist–Shannon sampling theorem solves, but in relation to information transmission channels of non-unit dimension. The paper proposes a method that gives possibility to determine the quantity of information recorded by wave oscillation receivers placed in a finite region of space. The methodological aspects of obtained results are discussed from the point of view of general information theory.